Notes : Specific Heat of Gas :Molar Specific Heat of a Gas at Contant volume and Pressure - Class 11 Physics Thermodynamics
Specific Heat Capacity: Definition, Formula, Units, Dimensions
Introduction
When a quantity of heat ($\Delta Q$) is supplied to a substance, its temperature changes from an initial temperature $T$ to a final temperature $T+\Delta T$.
Heat Capacity
Definition
Heat capacity is the amount of heat required to raise the temperature of a body by one kelvin.
Formula
$$S=\frac{\Delta Q}{\Delta T}$$
SI Unit
J K-1
Dimensions
$$[S]=[ML^2T^{-2}K^{-1}]$$
Specific Heat Capacity
Definition
Specific heat capacity is the amount of heat required to raise the temperature of unit mass of a substance by one kelvin.
Formula
$$s=\frac{\Delta Q}{m\Delta T}$$
$$s=\frac{S}{m}$$
$$\Delta Q=ms\Delta T$$
SI Unit
J kg-1 K-1
Dimensions
$$[s]=[L^2T^{-2}K^{-1}]$$
Molar Specific Heat Capacity
Definition
When the amount of substance is expressed in moles, the heat capacity per mole is called molar specific heat capacity.
Formula
$$C=\frac{\Delta Q}{n\Delta T}$$
$$C=\frac{S}{n}$$
SI Unit
J mol-1 K-1
Dimensions
$$[C]=[ML^2T^{-2}mol^{-1}K^{-1}]$$
Molar Specific Heat Capacity of Solids
Law of Equipartition of Energy
For one-dimensional vibration:
$$E=k_BT$$
For three-dimensional vibration:
$$E=3k_BT$$
For one mole of solid:
$$U=3k_BTN_A$$
Since
$$k_BN_A=R$$
Therefore
$$U=3RT$$
Dulong-Petit Law
$$\Delta V\approx0$$
$$\Delta Q\approx\Delta U$$
$$C=\frac{\Delta U}{\Delta T}=3R$$
$$C=24.94\;J\,mol^{-1}K^{-1}$$
Specific Heat Capacity of Water
Calorie
One calorie is the amount of heat required to raise the temperature of 1 gram of water by 1°C.
More precisely, one calorie is the heat required to raise the temperature of 1 gram of water from 14.5°C to 15.5°C.
Specific Heat of Water
$$s=4186\;J\,kg^{-1}K^{-1}$$
$$s=4.186\;J\,g^{-1}K^{-1}$$
Specific Heat Capacities of Gases
Specific Heat Definition
$$s=\frac{\Delta Q}{m\Delta T}$$
Molar Specific Heat Capacity
$$C=\frac{\Delta Q}{n\Delta T}$$
Molar Specific Heat at Constant Volume (Cv)
$$C_v=\left(\frac{\Delta Q}{\Delta T}\right)_v$$
Molar Specific Heat at Constant Pressure (Cp)
$$C_p=\left(\frac{\Delta Q}{\Delta T}\right)_p$$
$$C_p>C_v$$
Mayer's Relation
$$C_p-C_v=R$$
$$R=8.314\;J\,mol^{-1}K^{-1}$$
Dimensions of Gas Constant
$$[R]=[ML^2T^{-2}mol^{-1}K^{-1}]$$
Derivation of Mayer's Relation
$$\Delta Q=\Delta U+P\Delta V$$
At constant volume:
$$\Delta V=0$$
$$\Delta Q=\Delta U$$
$$C_v=\frac{\Delta U}{\Delta T}$$
At constant pressure:
$$C_p=\frac{\Delta U}{\Delta T}+P\left(\frac{\Delta V}{\Delta T}\right)_p$$
$$C_p=C_v+P\left(\frac{\Delta V}{\Delta T}\right)_p$$
Using the ideal gas equation:
$$PV=RT$$
$$P\left(\frac{\Delta V}{\Delta T}\right)_p=R$$
$$C_p=C_v+R$$
$$\boxed{C_p-C_v=R}$$
Frequently Asked Questions (FAQs)
What is specific heat capacity?
Specific heat capacity is the amount of heat required to raise the temperature of unit mass of a substance by one kelvin.
What is the SI unit of specific heat capacity?
J kg-1 K-1
Why is water's specific heat capacity high?
Due to strong hydrogen bonding between water molecules.
Why is Cp greater than Cv?
Because a gas performs expansion work at constant pressure.
MCQs
1. The SI unit of specific heat capacity is:
A) J kg-1
B) J K-1
C) J kg-1 K-1
D) J mol-1 K-1
Answer: C
2. The specific heat capacity of water is:
A) 418
B) 4186
C) 8400
D) 24.94
Answer: B
3. Mayer's relation is:
A) Cp + Cv = R
B) Cp - Cv = R
C) Cv - Cp = R
D) CpCv = R
Answer: B
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